- Email: [email protected]
Large amplitude electromagnetic waves in hot relativistic plasmas
Volume 45A, number 2
PHYSICS LETTERS
10 September 1973
LARGE AMPLITUDE ELECTROMAGNETIC WAVES IN HOT RELATIVISTIC PLASMAS K. SCHINDLER
and L. JANICKE
Ruhr-Universitrii Bochum, 463 Bochum, Germany Received 9 July 1973 In view of possible astrophysical applications the theory of large amplitude undamped electromagnetic waves is generalised to include thermal effects. Exact solutions of the non-linear relativistic Vlasov equations are constructed. Sufficiently large velocity spreads lead to large reductions in the cut-off frequency.
One of the motives for the study of large amplitude electromagnetic waves in relativistic plasmas is their potential capability of accelerating charged particles in the vicinity of pulsars up to cosmic ray energies [ 11. In that context plane waves with large amplitudes (v E eE,/m,uc 3 1, E, maximum electric field, u wave frequency, m, particle rest mass, e charge) have been discussed in the zero-temperature limit [2,3]. In the presence of thermalising processes (e.g. instabilities) a finite temperature theory seems to be necessary. Here we generalise the theory to a large class of particle distribution functions, obtaining exact solutions for arbitrary amplitudes and temperatures. We start from relativistic Vlasov theory [4] where the particle dynamics is characterised by a distribution function f for each particle species and the electromagnetic field by the y-component of the vector potential ‘17= Cd(t-z/&z)
A, =A(@ ,
.
,
lly ‘Py/m
0
c ,
f=F(IS(,II,,,M),Farbitrary The only equation equation
-a2A/az2
(I$ ‘P,/m
0
c)
,
.
(2)
left to solve is Maxwell’s current
= pojz -c-~
a2Alat2
(3)
with iz = -dp(A)ld
XjFJM2+(f12-
(1)
The wave propagates along z with constant phase velocity pC > c, polarisation oriented along y. The wave form A(n) is to be determined. Note that q is defined in such a way that A(r)) is periodic with 271. As discussed below, for the present purpose we can confine ourselves to the cases where the electrostatic potential vanishes exactly. Our method is based on the fact that the particle motion in the field (1) exhibits three explicit constants of motion IIx =Px/moc
with a = eA/m,c and P denoting generalised particle momentum. With the aid of these constants the general form of the distribution functions can be expressed as
l)(I+(Hy-
a)2t ilgt 1)
X dIIx dIIY d&f where ZZsums over the particle species. Integrating (3) one finds
K
~e,~~(1--1//3~)(dA/d~)~+p(A)= where K is a constant. Further integration
(4)
gives the solution
(5) The dispersion relation being
4llax Amh
s J&j
Jeou2(pz_1) CT w2
(6) 91
are the minimum and maximum where A,, and A,, values of A in the wave. For convenience we now Lorentz-transform to a new coordinate system ?? which moves with velocity c/P. In that frame there is no spatial dependence and a suitable set of the constants of motion is fi,, n,,, ?I,. To achieve consistency with the assumptions made (e.g. (p,&,Az = 0) we impose the condition that .I;lfix, l$,, ll,) be symmetric with respect to fi,, ii,,, ii,, and Ce
JFdiiX
diiYdiiz = 0 . cp,A,, A, vanish.
F - tqii,) G(rIy)@i,) we find from (5) and (6) a triangular wave form with the dispersion relation C
$
= 9,
we = plasma frequency
(7)
which was derived by Kennel et al. [3]. Simulating finite temperature effects by the choice F = %Q) we find
92
si$,
a) s(Q),
p=CmO;2 i(a-a)J_+((Wta)J_ Jl
+ln
t(a!-a)2-,tcu
[ Je2-a-a
D
where i? is the density of the particle species considered. If OLQ (al,,, for each species we recover (7). For
Then z, 2,) A,, and consequently Specialising for the cold case
c2
10 September 1973
PHYSICS LETTERS
Volume 45A, number 2
s=lforlii I (y Y
Clearly, thermal effects reduce the cut-off in a similar way as large amplitudes do in the cold case. The present approach can easily be generalised to situations with cpf 0, allowing for spatially varying wave forms without net bulk flow. Detials will be given elsewhere.
References [l] J.E. Gunn and J.P. Ostriker, Phys. Rev. Lett. 22 (1969) 128; see also H. Kegel, Astron. and Astrophys. 22 (1973) 475. [2] C. Max and F. Perkins, Phys. Rev. Lett. 27 (1971) 1342. [ 31 C.F. Kennel, G. Schmidt and T. Wilcox, Report PPG-144, University of California, Los Angles (1973). [4] 0. Buneman and W.B. Pardo, eds., Relativistic Plasmas (W.A. Benjamin, Inc., New York, 1968).
PHYSICS LETTERS
10 September 1973
LARGE AMPLITUDE ELECTROMAGNETIC WAVES IN HOT RELATIVISTIC PLASMAS K. SCHINDLER
and L. JANICKE
Ruhr-Universitrii Bochum, 463 Bochum, Germany Received 9 July 1973 In view of possible astrophysical applications the theory of large amplitude undamped electromagnetic waves is generalised to include thermal effects. Exact solutions of the non-linear relativistic Vlasov equations are constructed. Sufficiently large velocity spreads lead to large reductions in the cut-off frequency.
One of the motives for the study of large amplitude electromagnetic waves in relativistic plasmas is their potential capability of accelerating charged particles in the vicinity of pulsars up to cosmic ray energies [ 11. In that context plane waves with large amplitudes (v E eE,/m,uc 3 1, E, maximum electric field, u wave frequency, m, particle rest mass, e charge) have been discussed in the zero-temperature limit [2,3]. In the presence of thermalising processes (e.g. instabilities) a finite temperature theory seems to be necessary. Here we generalise the theory to a large class of particle distribution functions, obtaining exact solutions for arbitrary amplitudes and temperatures. We start from relativistic Vlasov theory [4] where the particle dynamics is characterised by a distribution function f for each particle species and the electromagnetic field by the y-component of the vector potential ‘17= Cd(t-z/&z)
A, =A(@ ,
.
,
lly ‘Py/m
0
c ,
f=F(IS(,II,,,M),Farbitrary The only equation equation
-a2A/az2
(I$ ‘P,/m
0
c)
,
.
(2)
left to solve is Maxwell’s current
= pojz -c-~
a2Alat2
(3)
with iz = -dp(A)ld
XjFJM2+(f12-
(1)
The wave propagates along z with constant phase velocity pC > c, polarisation oriented along y. The wave form A(n) is to be determined. Note that q is defined in such a way that A(r)) is periodic with 271. As discussed below, for the present purpose we can confine ourselves to the cases where the electrostatic potential vanishes exactly. Our method is based on the fact that the particle motion in the field (1) exhibits three explicit constants of motion IIx =Px/moc
with a = eA/m,c and P denoting generalised particle momentum. With the aid of these constants the general form of the distribution functions can be expressed as
l)(I+(Hy-
a)2t ilgt 1)
X dIIx dIIY d&f where ZZsums over the particle species. Integrating (3) one finds
K
~e,~~(1--1//3~)(dA/d~)~+p(A)= where K is a constant. Further integration
(4)
gives the solution
(5) The dispersion relation being
4llax Amh
s J&j
Jeou2(pz_1) CT w2
(6) 91
are the minimum and maximum where A,, and A,, values of A in the wave. For convenience we now Lorentz-transform to a new coordinate system ?? which moves with velocity c/P. In that frame there is no spatial dependence and a suitable set of the constants of motion is fi,, n,,, ?I,. To achieve consistency with the assumptions made (e.g. (p,&,Az = 0) we impose the condition that .I;lfix, l$,, ll,) be symmetric with respect to fi,, ii,,, ii,, and Ce
JFdiiX
diiYdiiz = 0 . cp,A,, A, vanish.
F - tqii,) G(rIy)@i,) we find from (5) and (6) a triangular wave form with the dispersion relation C
$
= 9,
we = plasma frequency
(7)
which was derived by Kennel et al. [3]. Simulating finite temperature effects by the choice F = %Q) we find
92
si$,
a) s(Q),
p=CmO;2 i(a-a)J_+((Wta)J_ Jl
+ln
t(a!-a)2-,tcu
[ Je2-a-a
D
where i? is the density of the particle species considered. If OLQ (al,,, for each species we recover (7). For
Then z, 2,) A,, and consequently Specialising for the cold case
c2
10 September 1973
PHYSICS LETTERS
Volume 45A, number 2
s=lforlii I
Clearly, thermal effects reduce the cut-off in a similar way as large amplitudes do in the cold case. The present approach can easily be generalised to situations with cpf 0, allowing for spatially varying wave forms without net bulk flow. Detials will be given elsewhere.
References [l] J.E. Gunn and J.P. Ostriker, Phys. Rev. Lett. 22 (1969) 128; see also H. Kegel, Astron. and Astrophys. 22 (1973) 475. [2] C. Max and F. Perkins, Phys. Rev. Lett. 27 (1971) 1342. [ 31 C.F. Kennel, G. Schmidt and T. Wilcox, Report PPG-144, University of California, Los Angles (1973). [4] 0. Buneman and W.B. Pardo, eds., Relativistic Plasmas (W.A. Benjamin, Inc., New York, 1968).